Question from measure theory Let $\Omega=[0,1]^2$ and let the algebra on it be the set of all rectangles and their finite unions.Let $f,g:[0,1]^2 \to \mathbb{R}$ be the projections onto the 1st and 2nd coordinates. On $\mathbb{R}$, the algebra under consideration is the collection of all intervals, together with their finite unions. Our goal is to show that $f$ and $g$ are measurable but their sum is not.
My question is twofold, 1) what does a rectangle mean in the context of $\Omega$? 
I assumed that it meant cartesian products of intervals, of the form $[x,y]\times [a,b]$ or  $(x,y)\times (a,b)$. This way it would be possible to show that $f$ and $g$ are measurable.
Now, my second question is, given my assumption about question 1, is my method of showing $f+g$ is not measurable correct?
Let us take $B=[\frac3 4,1]$. Now, $(f+g)^{-1}(B)$ is  $ \{ (x,y):\frac3 4 \leq x+y \leq 1 \} $. Now, to me, it is intuitive that this set cannot be written as a finite union of rectangles, as any rectangle containing the point $(1/2,1/2)$ would also contain either $(1/2,0)$ or $(0,1/2)$. Is this the right way to think?
 A: Question 1
Your guess is my initial guess too. However, if we really consider all rectangles, then we should also include rectangles that are rotated. Even so, I would probably still go with your first impression, that the rectangles' sides are parallel to the axes. If there's a way you can get clarification, it might be a good idea. (If it's homework, you can ask your teacher, and if it's from a book, see if they define "rectangle" anywhere.)
Question 2
You started out great but didn't finish well. Yes, we must show that $(f+g)^{-1}(B)$ is not in the algebra defined on $\Omega$. So, we must show that it isn't the finite union of rectangles.
About your approach, let me ask, what are the coordinates of the four corners of a square centered on $(1/2, 1/2)$, with sides of length 1/8? (Let's assume that the sides are parallel to the $x$ and $y$ axes.) And aren't the points $(1/2,0)$ and $(0,1/2)$ on the axes?
So how do we finish the proof?
It depends on the level of rigor required. At the very least, I would suggest drawing a picture of $A = (f+g)^{-1}(B)$. If the answer to question 1 is that the rectangles do indeed have sides parallel to the axes, then you might be able to just finish off with a simple geometric description. If full rigor is required, I'd do a proof by contradiction; I'd take an infinite sequence of points $\{x_n\}$ approaching one corner, say $x$. (Of course, we choose the sequence to be inside $A$). If we can write $A$ as a finite union of rectangles, then one rectangle has $x$ as a limit point. And therefore $\ldots$
(And don't forget to show that $f$ and $g$ are measurable.)
